Automorphisms of $\mathbb{C}^3$ Commuting with a $\mathbb{C}^+$-Action

TL;DR

This paper characterizes the group of polynomial automorphisms of a73 that commute with a given a7a7+ action, highlighting automorphisms from algebraic a7a7+ actions and their subgroup structure.

Contribution

It provides a detailed description of the centralizer of a a7a7+ action in a73, including the subgroup of automorphisms acting trivially on the quotient.

Findings

Automorphisms from algebraic a7a7+ actions are characterized.

The subgroup of automorphisms acting as the identity on the quotient is shown to be characteristic.

The structure of the centralizer a7(a7) is explicitly described.

Abstract

Let $\rho$ be an algebraic action of the additive group $\mathbb{C}^+$ on the three-dimensional affine space $\mathbb{C}^3$. We describe the group $\textrm{Cent}(\rho)$ of polynomial automorphisms of $\mathbb{C}^3$ that commute with $\rho$. A particular emphasis lies in the description of the automorphisms in $\textrm{Cent}(\rho)$ coming from algebraic $\mathbb{C}^+$-actions. As an application we prove that the automorphisms in $\textrm{Cent}(\rho)$ that are the identity on the algebraic quotient of $\rho$ form a characteristic subgroup of $\textrm{Cent}(\rho)$.